Let's try a few things to help determine the cause of the issue.
Let start by making sure you are using the current IP of your Azure SQL Database server. Let's ping the name of the Azure SQL server. The ping will fail but it should also return the current IP of the SQL Azure Server if DNS resolution is working.
C:\>ping <myserver>.database.windows.net
Pinging data.sn1-1.database.windows.net [65.55.74.144] with 32 bytes of data:
Request timed out.
Request timed out.
Request timed out.
Request timed out.
Ping statistics for 65.55.74.144:
Packets: Sent = 4, Received = 0, Lost = 4 (100% loss)
Now try to telnet to port 1433 using the IP address returned above. This article shows you how to enable Telnet on Windows.
C:\>telnet 65.55.74.144 1433
Connecting To 65.55.74.144...
If Telnet is successful, the window will change to a completely blank screen. Try connecting with that IP using SSMS again. If telenet is not successful, try tracert to identify at what step it fails to reach the Azure SQL Database server.
C:\>tracert 65.55.74.144
Tracing route to 65.55.74.144 over a maximum of 30 hops
1 2 ms 1 ms 1 ms 192.168.0.1
2 6 ms 5 ms 5 ms 10.33.34.50
3 5 ms 4 ms 5 ms 4-1-753.DLLSTX-LCR-07.verizon-gni.net [130.81.107.26]
4 8 ms 5 ms 5 ms so-6-2-0-0.DFW01-BB-RTR1.verizon-gni.net [130.81.28.208]
5 44 ms 43 ms 44 ms so-11-0-0-0.LCC1-RES-BB-RTR1-RE1.verizon-gni.net [130.81.17.40]
6 49 ms 44 ms 44 ms so-6-0-0-0.ASH-PEER-RTR1-re1.verizon-gni.net [130.81.10.90]